© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11n56 K11n58

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X5,15,6,14 X2837 X9,17,10,16 X11,19,12,18 X13,20,14,21 X15,7,16,6 X17,11,18,10 X19,1,20,22 X21,12,22,13

Gauss Code:

{1, -4, 2, -1, -3, 8, 4, -2, -5, 9, -6, 11, -7, 3, -8, 5, -9, 6, -10, 7, -11, 10}

DT (Dowker-Thistlethwaite) Code:

4 8 -14 2 -16 -18 -20 -6 -10 -22 -12

Alexander Polynomial:

- t-4 + 3t-3 - 2t-2 - t-1 + 3 - t - 2t2 + 3t3 - t4

Conway Polynomial:

1 + 2z2 - 4z4 - 5z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{7, 6}

Jones Polynomial:

q - q2 + 2q3 - q4 + q5 - q6

Other knots (up to mirrors) with the same Jones Polynomial:

{52, ...}

A2 (sl(3)) Invariant:

q4 + q6 + q8 + 2q10 + q12 - q16 - q18 - 2q20 + q26 + q32 - q34

HOMFLY-PT Polynomial:

- a-10 - a-10z2 + 4a-8 + 9a-8z2 + 6a-8z4 + a-8z6 - 7a-6 - 16a-6z2 - 16a-6z4 - 7a-6z6 - a-6z8 + 5a-4 + 10a-4z2 + 6a-4z4 + a-4z6

Kauffman Polynomial:

- a-11z + a-10 - 2a-10z2 - 3a-9z + 8a-9z3 - 6a-9z5 + a-9z7 + 4a-8 - 16a-8z2 + 24a-8z4 - 13a-8z6 + 2a-8z8 - 5a-7z + 7a-7z3 + 3a-7z5 - 5a-7z7 + a-7z9 + 7a-6 - 29a-6z2 + 40a-6z4 - 20a-6z6 + 3a-6z8 - 3a-5z - a-5z3 + 9a-5z5 - 6a-5z7 + a-5z9 + 5a-4 - 15a-4z2 + 16a-4z4 - 7a-4z6 + a-4z8

V2 and V3, the type 2 and 3 Vassiliev invariants:

{2, 3}

Khovanov Homology: (The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=6 is the signature of 1157. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 17               1 1 j = 15             1 1   j = 13           1 1 1   j = 11         1 2 1     j = 9       1 1         j = 7     1 1 1         j = 5   1 2             j = 3                   j = 1 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Show[DrawMorseLink[Knot[11, NonAlternating, 57]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 57]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 15, 6, 14], X[2, 8, 3, 7], > X[9, 17, 10, 16], X[11, 19, 12, 18], X[13, 20, 14, 21], X[15, 7, 16, 6], > X[17, 11, 18, 10], X[19, 1, 20, 22], X[21, 12, 22, 13]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 57]]
Out[4]=	GaussCode[1, -4, 2, -1, -3, 8, 4, -2, -5, 9, -6, 11, -7, 3, -8, 5, -9, 6, -10, > 7, -11, 10]
In[5]:=	DTCode[Knot[11, NonAlternating, 57]]
Out[5]=	DTCode[4, 8, -14, 2, -16, -18, -20, -6, -10, -22, -12]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 57]][t]
Out[6]=	-4 3 2 1 2 3 4 3 - t + -- - -- - - - t - 2 t + 3 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 57]][z]
Out[7]=	2 4 6 8 1 + 2 z - 4 z - 5 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 57]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 57]], KnotSignature[Knot[11, NonAlternating, 57]]}
Out[9]=	{7, 6}
In[10]:=	J=Jones[Knot[11, NonAlternating, 57]][q]
Out[10]=	2 3 4 5 6 q - q + 2 q - q + q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[5, 2], Knot[11, NonAlternating, 57]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 57]][q]
Out[12]=	4 6 8 10 12 16 18 20 26 32 34 q + q + q + 2 q + q - q - q - 2 q + q + q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 57]][a, z]
Out[13]=	2 2 2 2 4 4 4 6 -10 4 7 5 z 9 z 16 z 10 z 6 z 16 z 6 z z -a + -- - -- + -- - --- + ---- - ----- + ----- + ---- - ----- + ---- + -- - 8 6 4 10 8 6 4 8 6 4 8 a a a a a a a a a a a 6 6 8 7 z z z > ---- + -- - -- 6 4 6 a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 57]][a, z]
Out[14]=	2 2 2 2 -10 4 7 5 z 3 z 5 z 3 z 2 z 16 z 29 z 15 z a + -- + -- + -- - --- - --- - --- - --- - ---- - ----- - ----- - ----- + 8 6 4 11 9 7 5 10 8 6 4 a a a a a a a a a a a 3 3 3 4 4 4 5 5 5 6 8 z 7 z z 24 z 40 z 16 z 6 z 3 z 9 z 13 z > ---- + ---- - -- + ----- + ----- + ----- - ---- + ---- + ---- - ----- - 9 7 5 8 6 4 9 7 5 8 a a a a a a a a a a 6 6 7 7 7 8 8 8 9 9 20 z 7 z z 5 z 6 z 2 z 3 z z z z > ----- - ---- + -- - ---- - ---- + ---- + ---- + -- + -- + -- 6 4 9 7 5 8 6 4 7 5 a a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 57]], Vassiliev[3][Knot[11, NonAlternating, 57]]}
Out[15]=	{2, 3}
In[16]:=	Kh[Knot[11, NonAlternating, 57]][q, t]
Out[16]=	5 5 7 q q 7 9 7 2 9 2 11 2 11 3 2 q + q + -- + -- + q t + q t + q t + q t + q t + 2 q t + 2 t t 13 3 11 4 13 4 15 4 13 5 15 5 17 5 17 6 > q t + q t + q t + q t + q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n57

K11n56 K11n58